Japanese Language Stack Exchange is a question and answer site for students, teachers, and linguists wanting to discuss the finer points of the Japanese language. It's 100% free, no registration required.

Judging from the dictionary entries, these are basically the same, with つかまえる being in the first definition of とらえる. You may notice, however, that these words have alternate kanji, and the choice of kanji can affect the nuance. For example, definition 2 of つかまえる only applies to the other two kanji. So with this particular kanji the two words appear to be more or less the same.

Some question sites have some other interpretations from regular people, though, who have their own insights about the differences.

Basically the distinction is that つかまえる emphasizing that you use your hands or arms to catch something, whereas in とらえる the emphasis is on the capture, and the prevention of escape. とらえる is stronger than つかまえる. For example, you would つかまえる a bug or a small animal, but not とらえる. There's more explanation on that page too. It seems reliable to me.

Can you explain a bit about the use of J-J dictionaries? How did you understand from the definitions that there's more then one common kanji?
–
NescioNov 22 '12 at 7:39

@Nescio The markers/abbreviations used differ but there will be something called 凡例 which explains what the meanings are. Alternate kanji will be listed at the top of the entry, usually with markers that show if they are non-joyo or joyo with non-joyo readings. If there are multiple meanings/uses, some might have notes suggesting which kanji to use for which meaning. For example, if you look up みる, one meaning is something like "to examine", in a medical sense. This meaning has the note (「診る」とも書く) next to it, basically meaning 'is also written 診る.'
–
nkjtNov 22 '12 at 16:26